mixedcos

Average Error: 28.1 → 2.0

Time: 6.8s

Precision: binary64

\[ \begin{array}{c}[c, s] = \mathsf{sort}([c, s])\\ \end{array} \]

Math

\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

↓

\[\begin{array}{l} t_0 := \cos \left(x + x\right)\\ t_1 := c \cdot \left(x \cdot s\right)\\ t_2 := x \cdot \left(c \cdot s\right)\\ t_3 := \frac{1}{t_2}\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;t_0 \cdot \left(t_3 \cdot t_3\right)\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-253}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{t_1}^{2}}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-218}:\\ \;\;\;\;t_0 \cdot {\left(s \cdot \left(x \cdot c\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+217}:\\ \;\;\;\;t_0 \cdot {t_2}^{-2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {t_1}^{-2}\\ \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))

↓

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (+ x x)))
        (t_1 (* c (* x s)))
        (t_2 (* x (* c s)))
        (t_3 (/ 1.0 t_2)))
   (if (<= x -1.95e-62)
     (* t_0 (* t_3 t_3))
     (if (<= x -5.4e-253)
       (/ (cos (* x 2.0)) (pow t_1 2.0))
       (if (<= x 5.8e-218)
         (* t_0 (pow (* s (* x c)) -2.0))
         (if (<= x 1.85e+217)
           (* t_0 (pow t_2 -2.0))
           (* t_0 (pow t_1 -2.0))))))))

C

double code(double x, double c, double s) {
	return cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x));
}

↓

double code(double x, double c, double s) {
	double t_0 = cos((x + x));
	double t_1 = c * (x * s);
	double t_2 = x * (c * s);
	double t_3 = 1.0 / t_2;
	double tmp;
	if (x <= -1.95e-62) {
		tmp = t_0 * (t_3 * t_3);
	} else if (x <= -5.4e-253) {
		tmp = cos((x * 2.0)) / pow(t_1, 2.0);
	} else if (x <= 5.8e-218) {
		tmp = t_0 * pow((s * (x * c)), -2.0);
	} else if (x <= 1.85e+217) {
		tmp = t_0 * pow(t_2, -2.0);
	} else {
		tmp = t_0 * pow(t_1, -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))
end function

↓

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = cos((x + x))
    t_1 = c * (x * s)
    t_2 = x * (c * s)
    t_3 = 1.0d0 / t_2
    if (x <= (-1.95d-62)) then
        tmp = t_0 * (t_3 * t_3)
    else if (x <= (-5.4d-253)) then
        tmp = cos((x * 2.0d0)) / (t_1 ** 2.0d0)
    else if (x <= 5.8d-218) then
        tmp = t_0 * ((s * (x * c)) ** (-2.0d0))
    else if (x <= 1.85d+217) then
        tmp = t_0 * (t_2 ** (-2.0d0))
    else
        tmp = t_0 * (t_1 ** (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	return Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x));
}

↓

public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x + x));
	double t_1 = c * (x * s);
	double t_2 = x * (c * s);
	double t_3 = 1.0 / t_2;
	double tmp;
	if (x <= -1.95e-62) {
		tmp = t_0 * (t_3 * t_3);
	} else if (x <= -5.4e-253) {
		tmp = Math.cos((x * 2.0)) / Math.pow(t_1, 2.0);
	} else if (x <= 5.8e-218) {
		tmp = t_0 * Math.pow((s * (x * c)), -2.0);
	} else if (x <= 1.85e+217) {
		tmp = t_0 * Math.pow(t_2, -2.0);
	} else {
		tmp = t_0 * Math.pow(t_1, -2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	return math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))

↓

def code(x, c, s):
	t_0 = math.cos((x + x))
	t_1 = c * (x * s)
	t_2 = x * (c * s)
	t_3 = 1.0 / t_2
	tmp = 0
	if x <= -1.95e-62:
		tmp = t_0 * (t_3 * t_3)
	elif x <= -5.4e-253:
		tmp = math.cos((x * 2.0)) / math.pow(t_1, 2.0)
	elif x <= 5.8e-218:
		tmp = t_0 * math.pow((s * (x * c)), -2.0)
	elif x <= 1.85e+217:
		tmp = t_0 * math.pow(t_2, -2.0)
	else:
		tmp = t_0 * math.pow(t_1, -2.0)
	return tmp

Julia

function code(x, c, s)
	return Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x)))
end

↓

function code(x, c, s)
	t_0 = cos(Float64(x + x))
	t_1 = Float64(c * Float64(x * s))
	t_2 = Float64(x * Float64(c * s))
	t_3 = Float64(1.0 / t_2)
	tmp = 0.0
	if (x <= -1.95e-62)
		tmp = Float64(t_0 * Float64(t_3 * t_3));
	elseif (x <= -5.4e-253)
		tmp = Float64(cos(Float64(x * 2.0)) / (t_1 ^ 2.0));
	elseif (x <= 5.8e-218)
		tmp = Float64(t_0 * (Float64(s * Float64(x * c)) ^ -2.0));
	elseif (x <= 1.85e+217)
		tmp = Float64(t_0 * (t_2 ^ -2.0));
	else
		tmp = Float64(t_0 * (t_1 ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp = code(x, c, s)
	tmp = cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x));
end

↓

function tmp_2 = code(x, c, s)
	t_0 = cos((x + x));
	t_1 = c * (x * s);
	t_2 = x * (c * s);
	t_3 = 1.0 / t_2;
	tmp = 0.0;
	if (x <= -1.95e-62)
		tmp = t_0 * (t_3 * t_3);
	elseif (x <= -5.4e-253)
		tmp = cos((x * 2.0)) / (t_1 ^ 2.0);
	elseif (x <= 5.8e-218)
		tmp = t_0 * ((s * (x * c)) ^ -2.0);
	elseif (x <= 1.85e+217)
		tmp = t_0 * (t_2 ^ -2.0);
	else
		tmp = t_0 * (t_1 ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / t$95$2), $MachinePrecision]}, If[LessEqual[x, -1.95e-62], N[(t$95$0 * N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.4e-253], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-218], N[(t$95$0 * N[Power[N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.85e+217], N[(t$95$0 * N[Power[t$95$2, -2.0], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[t$95$1, -2.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Error

Bits error versus x

Bits error versus c

Bits error versus s

Derivation

1. Split input into 5 regimes

2. if x < -1.9500000000000002e-62

   1. Initial program 24.1

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. Simplified 2.0

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

   3. Applied egg-rr 2.3

      \[\leadsto \color{blue}{\cos \left(x + x\right) \cdot {\left(c \cdot \left(s \cdot x\right)\right)}^{-2}} \]

   4. Applied egg-rr 1.6

      \[\leadsto \cos \left(x + x\right) \cdot \color{blue}{\left(\frac{1}{x \cdot \left(c \cdot s\right)} \cdot \frac{1}{x \cdot \left(c \cdot s\right)}\right)} \]

   if -1.9500000000000002e-62 < x < -5.39999999999999998e-253

   1. Initial program 35.8

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. Simplified 2.8

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

   3. Applied egg-rr 4.3

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left({\left(c \cdot \left(s \cdot x\right)\right)}^{2}\right)}^{1}}} \]

   if -5.39999999999999998e-253 < x < 5.8000000000000004e-218

   1. Initial program 49.9

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. Simplified 18.0

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

   3. Applied egg-rr 4.5

      \[\leadsto \color{blue}{\cos \left(x + x\right) \cdot {\left(c \cdot \left(s \cdot x\right)\right)}^{-2}} \]

   4. Taylor expanded in c around 0 5.3

      \[\leadsto \cos \left(x + x\right) \cdot {\color{blue}{\left(s \cdot \left(c \cdot x\right)\right)}}^{-2} \]

   if 5.8000000000000004e-218 < x < 1.85000000000000005e217

   1. Initial program 26.2

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. Simplified 1.3

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

   3. Applied egg-rr 2.5

      \[\leadsto \color{blue}{\cos \left(x + x\right) \cdot {\left(c \cdot \left(s \cdot x\right)\right)}^{-2}} \]

   4. Applied egg-rr 0.9

      \[\leadsto \cos \left(x + x\right) \cdot {\color{blue}{\left({\left(x \cdot \left(c \cdot s\right)\right)}^{1}\right)}}^{-2} \]

   if 1.85000000000000005e217 < x

   1. Initial program 27.9

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. Simplified 5.6

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

   3. Applied egg-rr 2.6

      \[\leadsto \color{blue}{\cos \left(x + x\right) \cdot {\left(c \cdot \left(s \cdot x\right)\right)}^{-2}} \]

   4. Applied egg-rr 5.3

      \[\leadsto \cos \left(x + x\right) \cdot {\color{blue}{\left({\left(x \cdot \left(c \cdot s\right)\right)}^{1}\right)}}^{-2} \]

   5. Taylor expanded in x around 0 2.6

      \[\leadsto \cos \left(x + x\right) \cdot {\left({\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}}^{1}\right)}^{-2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;\cos \left(x + x\right) \cdot \left(\frac{1}{x \cdot \left(c \cdot s\right)} \cdot \frac{1}{x \cdot \left(c \cdot s\right)}\right)\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-253}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-218}:\\ \;\;\;\;\cos \left(x + x\right) \cdot {\left(s \cdot \left(x \cdot c\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+217}:\\ \;\;\;\;\cos \left(x + x\right) \cdot {\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(x + x\right) \cdot {\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))